oalg-base-1.1.4.0: Algebraic structures on oriented entities and limits as a tool kit to solve algebraic problems.
Copyright(c) Erich Gut
LicenseBSD3
Maintainerzerich.gut@gmail.com
Safe HaskellSafe-Inferred
LanguageHaskell2010

OAlg.Entity.Slice.Definition

Description

definition of slicing a Multiplicative structures according a given indexed Point.

Synopsis

Slice

data Slice s i c where Source #

slice over c by a given Site and indexed by i.

Constructors

SliceFrom :: i c -> c -> Slice From i c 
SliceTo :: i c -> c -> Slice To i c 

Instances

Instances details
(Show1 i, Show c) => Show (Slice s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

showsPrec :: Int -> Slice s i c -> ShowS #

show :: Slice s i c -> String #

showList :: [Slice s i c] -> ShowS #

(Eq1 i, Eq c) => Eq (Slice s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

(==) :: Slice s i c -> Slice s i c -> Bool #

(/=) :: Slice s i c -> Slice s i c -> Bool #

(Oriented c, Sliced i c) => Validable (Slice s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid :: Slice s i c -> Statement Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandard (Slice 'From i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandard :: X (Slice 'From i c) Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandard (Slice 'To i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandard :: X (Slice 'To i c) Source #

(Oriented c, Sliced i c, Typeable s) => Entity (Slice s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type Dual (Slice s i c :: Type) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type Dual (Slice s i c :: Type) = Slice (Dual s) i (Op c)

slice :: Slice s i c -> c Source #

the underlying slice.

Factor

data SliceFactor s i c Source #

factor between two slices.

Property Let SliceFactor a b f be in SliceFactor s i c for a Multiplicative structure c constrained by Sliced i c then holds:

  1. If a matches SliceFrom _ a' then holds: Let SliceFrom _ b' = b in

    1. orientation f == end a' :> end b'.
    2. b' == f * a'.

  2. If a matches SliceTo _ a' then holds: Let SliceTo _ b' = b in

    1. orientation f == start a' :> start b'.
    2. a' == b' * f .

Constructors

SliceFactor (Slice s i c) (Slice s i c) c 

Instances

Instances details
(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardOrtSite 'From (SliceFactor 'From i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandardOrtSite :: XOrtSite 'From (SliceFactor 'From i c) Source #

XStandardOrtSite 'From (SliceFactor 'To (Proxy :: Type -> Type) OS) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandardOrtSite :: XOrtSite 'From (SliceFactor 'To Proxy OS) Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardOrtSite 'To (SliceFactor 'To i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandardOrtSite :: XOrtSite 'To (SliceFactor 'To i c) Source #

(Show1 i, Show c) => Show (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

showsPrec :: Int -> SliceFactor s i c -> ShowS #

show :: SliceFactor s i c -> String #

showList :: [SliceFactor s i c] -> ShowS #

(Eq1 i, Eq c) => Eq (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

(==) :: SliceFactor s i c -> SliceFactor s i c -> Bool #

(/=) :: SliceFactor s i c -> SliceFactor s i c -> Bool #

(Multiplicative c, Sliced i c) => Validable (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid :: SliceFactor s i c -> Statement Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandard (SliceFactor 'From i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandard :: X (SliceFactor 'From i c) Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandard (SliceFactor 'To i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

xStandard :: X (SliceFactor 'To i c) Source #

(Multiplicative c, Sliced i c, Typeable s) => Entity (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

(Multiplicative c, Sliced i c, Typeable s) => Multiplicative (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

one :: Point (SliceFactor s i c) -> SliceFactor s i c Source #

(*) :: SliceFactor s i c -> SliceFactor s i c -> SliceFactor s i c Source #

npower :: SliceFactor s i c -> N -> SliceFactor s i c Source #

(Multiplicative c, Sliced i c, Typeable s) => Oriented (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Associated Types

type Point (SliceFactor s i c) Source #

Methods

orientation :: SliceFactor s i c -> Orientation (Point (SliceFactor s i c)) Source #

start :: SliceFactor s i c -> Point (SliceFactor s i c) Source #

end :: SliceFactor s i c -> Point (SliceFactor s i c) Source #

(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardOrtSiteFrom (SliceFactor 'From i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

XStandardOrtSiteFrom (SliceFactor 'To (Proxy :: Type -> Type) OS) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardOrtSiteTo (SliceFactor 'To i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardPoint (SliceFactor 'From i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardPoint (SliceFactor 'To i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type Dual (SliceFactor s i c :: Type) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type Dual (SliceFactor s i c :: Type) = SliceFactor (Dual s) i (Op c)
type Point (SliceFactor s i c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type Point (SliceFactor s i c) = Slice s i c

slfDrop :: SliceFactor s i c -> c Source #

the underlying factor.

Sliced

class (Entity1 i, Singleton1 i) => Sliced i c where Source #

Slicing a Multiplicative structures at the Point given by the type of the index i.

Note The constraint Singleton1 i ensures that the distinguished point depends only on the type i c.

Methods

slicePoint :: i c -> Point c Source #

the distingueished point of the given index type i.

Instances

Instances details
Sliced i c => Sliced i (Op c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

slicePoint :: i (Op c) -> Point (Op c) Source #

Sliced (Proxy :: Type -> Type) OS Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

slicePoint :: Proxy OS -> Point OS Source #

Hom

data SliceFactorDrop s x y where Source #

dropping a slice factor.

Constructors

SliceFactorFromDrop :: (Multiplicative c, Sliced i c) => SliceFactorDrop From (SliceFactor From i c) c 
SliceFactorToDrop :: (Multiplicative c, Sliced i c) => SliceFactorDrop To (SliceFactor To i c) c 

Instances

Instances details
Applicative (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

amap :: SliceFactorDrop s a b -> a -> b Source #

EmbeddableMorphismTyp (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Morphism (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Associated Types

type ObjectClass (SliceFactorDrop s) Source #

Methods

homomorphous :: SliceFactorDrop s x y -> Homomorphous (ObjectClass (SliceFactorDrop s)) x y Source #

domain :: SliceFactorDrop s x y -> Struct (ObjectClass (SliceFactorDrop s)) x Source #

range :: SliceFactorDrop s x y -> Struct (ObjectClass (SliceFactorDrop s)) y Source #

Eq2 (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

eq2 :: SliceFactorDrop s x y -> SliceFactorDrop s x y -> Bool Source #

Show2 (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

show2 :: SliceFactorDrop s a b -> String Source #

Validable2 (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid2 :: SliceFactorDrop s x y -> Statement Source #

Typeable s => Entity2 (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Typeable s => HomMultiplicative (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Typeable s => HomOriented (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

pmap :: SliceFactorDrop s a b -> Point a -> Point b Source #

EmbeddableMorphism (SliceFactorDrop s) Typ Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

EmbeddableMorphism (SliceFactorDrop s) Mlt Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

EmbeddableMorphism (SliceFactorDrop s) Ort Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Show (SliceFactorDrop s x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

showsPrec :: Int -> SliceFactorDrop s x y -> ShowS #

show :: SliceFactorDrop s x y -> String #

showList :: [SliceFactorDrop s x y] -> ShowS #

Eq (SliceFactorDrop s x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

(==) :: SliceFactorDrop s x y -> SliceFactorDrop s x y -> Bool #

(/=) :: SliceFactorDrop s x y -> SliceFactorDrop s x y -> Bool #

Validable (SliceFactorDrop s x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid :: SliceFactorDrop s x y -> Statement Source #

(Typeable s, Typeable x, Typeable y) => Entity (SliceFactorDrop s x y) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type ObjectClass (SliceFactorDrop s) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

type ObjectClass (SliceFactorDrop s) = Mlt

Limes

data DiagramSlicedCenter i t n m c where Source #

predicate for a Star t diagram with center Point given by the index type i c.

Property Let DiagramSlicedCenter i d be in DiagramSlicedCenter i t n m c then holds: slicePoint i == dgCenter d.

Constructors

DiagramSlicedCenter :: Sliced i c => i c -> Diagram (Star t) n m c -> DiagramSlicedCenter i t n m c 

Instances

Instances details
Oriented c => Show (DiagramSlicedCenter i t n m c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

showsPrec :: Int -> DiagramSlicedCenter i t n m c -> ShowS #

show :: DiagramSlicedCenter i t n m c -> String #

showList :: [DiagramSlicedCenter i t n m c] -> ShowS #

Oriented c => Validable (DiagramSlicedCenter i t n m c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid :: DiagramSlicedCenter i t n m c -> Statement Source #

data LimesSlicedTip i s p t n m c where Source #

predicate for a limes with a sliced tip of the universal cone.

Property Let LimesSlicedTip i l be in LimesSlicedTip i s p t n m c then holds: tip (universalCone l) == slicePoint i.

Constructors

LimesSlicedTip :: Sliced i c => i c -> Limes s p t n m c -> LimesSlicedTip i s p t n m c 

Instances

Instances details
Oriented c => Show (LimesSlicedTip i s p t n m c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

showsPrec :: Int -> LimesSlicedTip i s p t n m c -> ShowS #

show :: LimesSlicedTip i s p t n m c -> String #

showList :: [LimesSlicedTip i s p t n m c] -> ShowS #

(Oriented c, Validable (Limes s p t n m c)) => Validable (LimesSlicedTip i s p t n m c) Source # 
Instance details

Defined in OAlg.Entity.Slice.Definition

Methods

valid :: LimesSlicedTip i s p t n m c -> Statement Source #

lstLimes :: LimesSlicedTip i s p t n m c -> Limes s p t n m c Source #

the underlying limes.

Projective

slfTerminalPoint :: (Multiplicative c, Sliced i c) => TerminalPoint (SliceFactor To i c) Source #

terminal point for factors sliced to a Point.

slfPullback :: Multiplicative c => Products n (SliceFactor To i c) -> DiagramSlicedCenter i To (n + 1) n c -> Pullback n c Source #

the induced pullback.

Injective

slfLimitsInjective :: (Multiplicative c, Sliced i c) => Limits Mlt Injective t n m c -> Limits Mlt Injective t n m (SliceFactor To i c) Source #

the induced Injective Limits.

X

xSliceTo :: Sliced i c => XOrtSite To c -> i c -> X (Slice To i c) Source #

the induced random variable.

xSliceFrom :: Sliced i c => XOrtSite From c -> i c -> X (Slice From i c) Source #

the induced random variable.

xosXOrtSiteToSliceFactorTo :: (Multiplicative c, Sliced i c) => XOrtSite To c -> i c -> XOrtSite To (SliceFactor To i c) Source #

the induced random variable.

xosXOrtSiteFromSliceFactorFrom :: (Multiplicative c, Sliced i c) => XOrtSite From c -> i c -> XOrtSite From (SliceFactor From i c) Source #

the induced random variable.